Let's go through each question one by one and solve them.

Question 1

You are asked to simplify the expression:

$(\sqrt{5} + 3\sqrt{2})(\sqrt{5} - 2\sqrt{2})$

This is a binomial product, and we can expand it using the distributive property (FOIL method):

$(\sqrt{5} + 3\sqrt{2})(\sqrt{5} - 2\sqrt{2}) = \sqrt{5} \cdot \sqrt{5} + \sqrt{5} \cdot (-2\sqrt{2}) + 3\sqrt{2} \cdot \sqrt{5} + 3\sqrt{2} \cdot (-2\sqrt{2})$

Now let's simplify each term:

1. $\sqrt{5} \cdot \sqrt{5} = 5$
2. $\sqrt{5} \cdot (-2\sqrt{2}) = -2\sqrt{10}$
3. $3\sqrt{2} \cdot \sqrt{5} = 3\sqrt{10}$
4. $3\sqrt{2} \cdot (-2\sqrt{2}) = -6 \cdot 2 = -12$

Now, add all these terms together:

$5 + (-2\sqrt{10}) + 3\sqrt{10} + (-12)$

Combine like terms:

$5 - 12 = -7$ $-2\sqrt{10} + 3\sqrt{10} = \sqrt{10}$

Thus, the expression simplifies to:

$\boxed{\sqrt{10} - 7}$

This corresponds to Option C.

Question 2

Simplify the radical expression:

$\sqrt[3]{x^{15}}$

The property of radicals and exponents states that:

$\sqrt[n]{x^m} = x^{m/n}$

So in this case:

$\sqrt[3]{x^{15}} = x^{15/3} = x^5$

Thus, the simplified expression is:

$\boxed{x^5}$

This corresponds to Option D.

Do you want more details or have any questions?

Here are 5 related questions for practice:

1. Simplify $(\sqrt{7} + 2\sqrt{3})(\sqrt{7} - 4\sqrt{3})$.
2. Simplify $(3\sqrt{5} + \sqrt{2})(3\sqrt{5} - \sqrt{2})$.
3. Simplify $\sqrt[4]{x^8}$.
4. Simplify $(2\sqrt{3} + 5)(2\sqrt{3} - 5)$.
5. Simplify $\sqrt[6]{x^{12}}$.

Tip: When simplifying expressions involving radicals, always look for patterns like the difference of squares or the application of exponent rules to reduce the expression.